1. Field of the Invention
Embodiments of the present invention generally relate to actuators and more specifically to tailoring the moments of inertia of at least two components, of the actuator, to enhance or suppress bending.
2. Description of the Related Art
It has long been known that two sheets of metal with different coefficients of thermal expansion (“CTE”) will bend with changes in temperature. The traditional approach to this technology is illuminated by the “bi-metallic spring.” The general relationship for when the two metals (i.e., in a bi-metallic strip) are of the same thickness is analyzed in Timoshenko, S., Analysis of Bi-metal Thermostats, J. Opt. Soc. Am. (1925), 11(2), pp. 233-255 (hereinafter “Timoshenko”).
Timoshenko analyzed the bending of a bi-metal thermostat of rectangular cross-section and concluded:                The curvature is proportional to the difference in elongation of the two metals and inversely proportional to the thickness of the strip. It is seen that the magnitude of the ratio [of the Young's moduli of the two metals] does not produce any substantial effect on the curvature of the strip. See Timoshenko at page 235.        
A bi-metallic strip 100 is provided in FIG. 1. Specifically, the bi-metallic strip 100 includes a first component 102 and a second component 104. The first component 102 is made of a different metal than the second component 104. The first component 102 and second component 104 have the same dimensions and different coefficients of linear expansion and moments of inertia. Heating the bi-metallic strip 100 causes bending of the bi-metallic strip 100.
Equations are provided below for calculating the temperature of buckling, the complete travel during buckling, and the temperature of buckling in a backward direction. By using these equations, the dimensions of the bi-metallic strip 100 for a given temperature of operation and a given complete range of temperature can be calculated. It has long been known that two sheets of metal with different coefficients of thermal expansion (“CTE”) will bend with changes in temperature. The general relationship for the two metals (i.e., components 102 and 104) are of the same thickness (as provided by Timoshenko) is provided by Equation 1:
                              1          ρ                =                              24            ⁢                          (                                                α                  2                                -                                  α                  1                                            )                        ⁢                          (                              Δ                ⁢                                                                  ⁢                T                            )                                            h            ⁡                          (                              14                +                n                +                                  1                  n                                            )                                                          Equation        ⁢                                  ⁢                  (          1          )                    
where the CTEs of the two materials are α1 (component 102) and α2 (component 104), the change in temperature is ΔT, h is the combined thickness of components 102 and 104, n is the ratio of the mechanical moduli of components 102 and 104, the radius of curvature is ρ, and the “curvature” is 1/ρ.
Note that Equation (1) can alternatively be expressed as Equation (2) below.
                              1                      ρ            rect                          =                              6            ⁢                          (                                                α                  2                                -                                  α                  1                                            )                        ⁢                          (                              t                -                                  t                  0                                            )                        ⁢                                          (                                  1                  +                  m                                )                            2                                            h            ⁡                          [                                                3                  ⁢                                                            (                                              1                        +                        m                                            )                                        2                                                  +                                                      (                                          1                      +                                              n                        ⁢                                                                                                  ⁢                        m                                                              )                                    ⁢                                      (                                                                  m                        2                                            +                                              1                                                  n                          ⁢                                                                                                          ⁢                          m                                                                                      )                                                              ]                                                          Equation        ⁢                                  ⁢                  (          2          )                    
where
  1      ρ    rect  is the radius of curvature of the strip 100, h is the height or diameter of the fiber, α2 is the coefficient of thermal expansion for component 104, α1 is the coefficient of thermal expansion for component 102, n is the ratio of the Young's moduli of the components 102 and 104, and m is a ratio of the thickness of components 102 and 104. Note that setting m=1 in Equation (2) yields Equation (1).
According to the analysis in Timoshenko, the bending of the bonded metal sheets (i.e., components 102 and 104 combined) is not a strong function of the mechanical modulus of the component metals. E1 and E2 are the elastic moduli of components 102 and 104, respectively. It is seen that the magnitude of
  n  =            E      1              E      2      does not produce any substantial effect on the curvature of the strip. For example, when n=1, then Equation (1) above is reduced to Equation (3).
                              1          ρ                =                              3            ⁢                          (                                                α                  2                                -                                  α                  1                                            )                        ⁢                          (                              Δ                ⁢                                                                  ⁢                T                            )                                            2            ⁢            h                                              Equation        ⁢                                  ⁢                  (          3          )                    
where the CTEs of the two materials are α1 (component 102) and α2 (component 104), the change in temperature is ΔT, h is the combined thickness of components 102 and 104, the radius of curvature is ρ, and the “curvature” is 1/ρ.
Similarly, when n=½ or n=2 then Equation (1) is reduced to Equation (4) below.
                              1          ρ                =                              48            ⁢                          (                                                α                  2                                -                                  α                  1                                            )                        ⁢                          (                              Δ                ⁢                                                                  ⁢                T                            )                                            33            ⁢            h                                              Equation        ⁢                                  ⁢                  (          4          )                    
where the CTEs of the two materials are α1 (component 102) and α2 (component 104), the change in temperature is ΔT, h is the combined thickness of components 102 and 104, the radius of curvature is ρ, and the “curvature” is 1/ρ.
“For a ratio of Young's moduli of 2, the “difference . . . [in curvature] is only about 3 percent.” See Timoshenko at page 236.
However, many combinations of materials that could be useful have mechanical moduli which can vary by a factor or ten or more. Based on these same equations the bending would be reduced by about one third. If the mechanical moduli differ by two orders of magnitude the bending is reduced to only 15% of the amount of bending that would be seen in the case where mechanical moduli are equal.
Examples of materials with very different mechanical moduli are polymers above and below their glass transition temperatures. Amorphous polymers above their glass transition temperature (i.e., in a rubbery state) usually have much higher CTEs than those below their glass transition temperature (i.e., in a glassy state) and would make good candidates for bending in response to temperature changes to act as a thermostat or a temperature adaptive insulation. Unfortunately, a decrease in modulus of about 3 orders of magnitude occurs at the glass transition making this combination of materials essentially useless.
Many materials of current technological interest such as gels, amorphous metals, shape memory polymers, and nanocomposites have mechanical moduli which vary by orders of magnitude limiting the combinations of materials that can be used.
However, for polymeric materials, the elastic modulus can change by three orders of magnitude below and above the glass transition temperature (Aklonis and McKnight, 1983). It is just such a combination of a polymer above its glass transition temperature and one below its glass transition temperature (or in a crystalline form) that gives the greatest difference in coefficients of thermal expansion.
The prior art (e.g., U.S. Pat. No. 4,115,620 issued Sep. 19, 1978) discloses an even polymer blend (i.e., extruded at 50:50 ratio).
Although U.S. Pat. No. 4,315,881 (issued Feb. 16, 1982) discloses that the ratio by weight of extruded fiber components is 30:70 to 70:30 and 40:60 to 60:40.
Generally, the prior art does not use the moments of inertia of the components to determine the ratio of those components (in an extruded material) and manipulate the shape of the extruded material to maximize bending.
Thus there is a need to use a wider selection of materials of significantly different mechanical moduli. There is also a need to tailor the moments of inertia of at least two components, in an actuator, to enhance or suppress bending.